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Solving Two-Trust-Region Subproblems Using Semidefinite Optimization with Eigenvector Branching

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Abstract

Semidefinite programming (SDP) problems typically utilize a constraint of the form \(X\succeq xx^T\) to obtain a convex relaxation of the condition \(X=xx^T\), where \(x\in \mathbb {R}^n\). In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of \(X-xx^T\). This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383–411, 2010, https://doi.org/10.1007/s10107-010-0371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem.

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Notes

  1. Data for these problem instances are available from the author on request.

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Acknowledgements

I am grateful to Sam Burer for helpful discussions on the topic of this paper and to Marco Locatelli for providing details of the computational results in [13]. I would also like to thank two anonymous referees for their careful readings of the paper and suggestions for improvements.

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  1. Department of Business Analytics, University of Iowa, Iowa City, IA, 52242, USA

    Kurt M. Anstreicher

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Correspondence to Kurt M. Anstreicher.

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Communicated by Etienne de Klerk.

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Anstreicher, K.M. Solving Two-Trust-Region Subproblems Using Semidefinite Optimization with Eigenvector Branching. J Optim Theory Appl 202, 303–319 (2024). https://doi.org/10.1007/s10957-022-02064-5

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